The Nielsen numbers of virtually unipotent maps on infra-nilmanifolds (Q2703552)

Nielsen number \(N(f)\) and Lefschetz number \(L(f)\) are two homotopy invariants to provide information on the fixed-point set of a self-map \(f\). An interesting question is to find relations between the two numbers. \textit{D. Anosov} showed that \(N(f) =|L(f)|\) for all maps on nilmanifolds. This result was generalized by \textit{E. C. Keppelmann} and \textit{C. K. McCord} [Pac. J. Math. 170, No. 1, 143-159 (1995; Zbl 0856.55003)] to a large class of solvmanifolds. But, the equality \(N(f) =|L(f)|\) can not be extended to infra-nilmanifolds. On the other hand, \textit{S. Kwasik} and \textit{K. B. Lee} [J. Lond. Math. Soc., II. Ser. 38, No. 3, 544-554 (1988; Zbl 0675.55004)] show that for homotopically periodic self-maps on infra-nilmanifolds \(N(f) =L(f)\). The author of this paper shows that for more general maps on infra-nilmanifolds, more precisely for maps in the homotopy class of the affine diffeomorphism which is said to be virtually unipotent, the equality \(N(f) =L(f)\) still holds. The key theorem (4.3) of this paper is based on the result of \textit{K. B. Lee} [Pac. J. Math. 168, No. 1, 157-166 (1995; Zbl 0920.55003)].